Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

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Enstrophy dissipation for two-dimensional incompressible flows

Dr. Anna Mazzucato

Department of Mathematics at Penn State University

Abstract:   We consider the problem of enstrophy dissipation for two-dimensional incompressible flows at high Reynolds number. We discuss two notions of enstrophy defects associated to approximate solution sequences to the Euler equations obtained by mollification and by vanishing viscosity. These notions were originally introduced by G. Eyink in order to reconcile the Kraichnan-Batchelor theory of 2D turbulence with properties of weak solutions to 2D Euler. We show that if the initial enstrophy is finite, the viscous enstrophy defect is well-defined. If the initial vorticity belongs to certain logarithmic refinements of L^2, then an exact transport equation holds for the corresponding enstrophy density, because of cancellation properties of the Biot-Savart kernel. Moreover, the transport defect also exists. For rougher data in a Besov space, for which the initial enstrophy is infinite, the enstrophy defects depend upon the approximating sequence and we produce examples of solutions where true dissipation occurs. This is joint work with Helena and Milton Lopes.

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